EP1372000B1 - Method for detection as well as for estimation of velocity and position of moving objects in SAR images - Google Patents

Abstract

One or more velocity components of a detected candidate (target) are estimated together with one or more of its positional components. The position of the candidate in individual images of the sequence from single-look synthetic aperture radar (SAR), is taken into account, together with the overhead for estimation of the required components. The overhead is minimized.

Description

The invention relates to a method for detection and velocity and position estimation of moving objects according to the preamble of patent claim 1.

State of the art

• Bei radialer Bewegung entsteht eine zusätzliche Dopplerverschiebung, die in einem Versatz des Objekts in positiver oder negativer Azimutrichtung, also in bzw. entgegengesetzt zur Flugrichtung der Radar-Antenne, resultiert. Diese Bewegung geht einher mit einer Geschwindigkeitskomponente in Rangerichtung (auch Entfernungsrichtung oder Cross-Track-Richtung), so dass diese Geschwindigkeitskomponente im einfach Rangegeschwindigkeit genannt wird.
• Bei großer Rangegeschwindigkeit verschwindet das Objekt im SAR-Bild.
• Bei Bewegung des Objekts in Azimutrichtung (Flugrichtung) verändert sich die Dopplerrate, wodurch das Objekt im SAR-Bild in Azimutrichtung verzerrt wird. In diesem Fall ist also eine Azimutkomponente (Along-track-Komponente) der Objektgeschwindigkeit vorhanden, die im kurz Azimutgeschwindigkeit genannt wird.

When obtaining images from SAR (Synthetic Aperture Radar) raw data, aberrations occur when moving objects (eg vehicles, also called moving targets) are in the scene. Depending on the direction of movement of the objects there are three main effects [1]:

• In the case of radial movement, an additional Doppler shift results, which results in an offset of the object in a positive or negative azimuth direction, ie in or opposite to the direction of flight of the radar antenna. This movement is accompanied by a velocity component in directional direction (also distance direction or cross-track direction), so that this velocity component is called in simple range speed.
• At high range speed, the object disappears in the SAR image.
• Moving the object in the azimuth direction (direction of flight) changes the Doppler rate, distorting the object in the azimuth SAR image. In this case, therefore, there is an azimuth component (along-track component) of the object speed, which is called azimuth speed in short.

Further effects occur when the moving object is accelerated but should not be considered here. In order to eliminate the described aberrations, it is necessary to detect the moving objects and estimate their speed and position. Also advantageous is the estimation of the shape of the objects.

For the detection of static objects in SAR images, a method is known from the literature which exploits the covariance matrix between the pixels of differently polarized images for detection [2]. However, images with different polarizations are not always available.

Most Moving Target Indication (MTI) methods known in the literature work with range-compressed SAR data, either in the Doppler frequency domain (e.g., [3], [4]) or in time Frequency level [5] are transformed. These methods are unable to detect moving objects that move only in the azimuth direction, since such objects are filtered out along with the background data (clutter data). In addition, this method requires a pulse repetition frequency (PRF) that is much larger than the fixed background Doppler bandwidth (clutter bandwidth) and therefore imposes restrictions on the usable SAR system. Finally, these methods result in some cases in large estimation inaccuracies.

DE 44 23 899 C1 and [6] also proposes a method for detection, localization and velocity estimation of moving targets from raw SAR data operating in the frequency domain. In this case, the correlation between two temporally successively formed azimuth spectra is exploited. This method is capable of estimating the azimuth velocity of moving objects, but has drawbacks with regard to the representation of the moving sequences of the moving objects. In particular, a direct assignment of the obtained velocity values to the moving objects in the SAR image is not possible. In addition, only objects are detected whose azimuth speed is greater than 22 km / h. The range speed is estimated only very roughly.

In DE 196 18 155 C2 and [7], a method is presented which exploits the temporal correlation in a sequence of single-look SAR images (SL images) for the detection and velocity estimation of moving objects. This sequence is first generated from range-compressed data using the entire available Doppler spectrum. After the generation of the sequence, a detection of candidates for moving objects in the single-look SAR images by searching for regions with respect to the environment deviating course of intensity takes place. The subsequent velocity estimate of the detected candidate is made by estimating the displacement vector of a candidate using the correlation between successive single-look SAR images.
With the method known from DE 196 18 155 C2, however, it is not possible to detect objects which move exclusively in the direction of the ranger. Another disadvantage is that the measuring range of the range speed is limited by the PRF specified by the system. In addition, no estimate of the position of the moving object is described in DE 196 18 155 C2.

It is therefore the object of the invention to provide a method with which an independent of the direction of movement of the objects detection at extended Geschwindigkeitsmeßbereich and a position estimate is possible.

This object is achieved by the method according to claim 1. Advantageous embodiments of the invention are the subject of further claims.

According to the invention, the estimation of one or more velocity components of a detected candidate is carried out together with the estimation of one or more positional components of the candidate, taking into account the positions of the candidate in the individual single-look SAR images of the sequence, a cost function being dependent on the parameters to be estimated , which is minimized by means of an optimization method.

For the inventive method, only data of a conventional SAR system are assumed, ie a system with a polarization, a carrier frequency and a channel. With the method according to the invention can be arbitrary On the Earth's surface moving objects are detected and their azimuth and range speeds and their position can be estimated with good accuracy. This method uses a PRF that is only slightly larger than the clutter bandwidth.

With the method according to the invention is a compensation of aberrations, d. H. a compensation of all three mentioned effects, and thus a distortion-free representation of the moving objects possible. The results of the method according to the invention can be visualized by displaying the SL images as a moving picture sequence with a resting background and marked, moving objects.

Fig. 1

Generierung einer Sequenz von Single-Look SAR-Bildern a) mit Darstellung der Range-Migration-Hyperbel eines statischen Objekts in der Range-Doppler-Ebene vor der Range-Migration-Korrektur (unten) und b) nach Range-Migration-Korrektur

Fig. 2

Messung eines Verschiebungsvektors (Displacementvektors) aus einem Bildpaar der Sequenz (Blockmatching)

Fig. 3

Zusammenhang zwischen der in den SL-Bildern zu beobachtenden Azimutgeschwindigkeit v_Mx,a und der wirklichen Geschwindigkeit v_Mx in Azimutrichtung ohne Berücksichtigung von Störungen

Fig. 4

Beobachtungsmodell zur unabhängigen Schätzung der Azimutgeschwindigkeit. Gemessen werden die am Ausgang anliegenden durch Rauschen gestörten Komponenten v_Mx,m ^(k).

Fig. 5

Beobachtungsmodell zur unabhängigen Schätzung der Rangegeschwindigkeit

Fig. 6

Dopplerspektren für jeweils ein statisches und ein bewegtes Objekt; Die Objektbewegung in Rangerichtung und die daraus resultierende Doppler- Verschiebung f_M ist so groß, dass Aliasing im Dopplerspektrum entsteht (Abtastfrequenz ist die PRF f_p); a) Fall a=1 bzw. 0 (positive Rangegeschwindigkeit); b) Fall a=-1 bzw. 0 (negative Rangegeschwindigkeit);

Fig. 7

Vergleich der Detektions- und Meßbereiche der Objektgeschwindigkeit für verschiedene Verfahren (bezogen auf DOSAR-Parameter, ein Kanal, C-Band)

Fig. 8

Schätzergebnisse für Azimut- und Rangegeschwindigkeit; Dargestellt ist jeweils die Standardabweichung des Schätzfehlers

Fig. 9

Schätzergebnisse für Azimut- und Rangeposition; Dargestellt ist jeweils die Standardabweichung des Schätzfehlers;

Fig. 10 a)

Single-Look SAR-Bild Nr. 17 aus einer generierten SAR-Bildsequenz, bestehend aus 30 Bildern; Lookmittenfrequenz 75 Hz; Die bewegten Objekte erscheinen verzerrt (vor allem unteres Objekt) und abseits von der Straße bzw. teilweise sind sie ganz verschwunden;

Fig. 10 b)

Kompensiertes Single-Look SAR-Bild aus der kompensierten SAR-Bildsequenz. Die geschätzte Bewegungsrichtung der detektierten bewegten Objekte ist mit einem Pfeil gekennzeichnet.

Fig. 11 a)

Multi-Look-Bild eines konventionellen SAR-Prozessors. Die bewegten Objekte erscheinen verzerrt und in Azimutrichtung verschoben. Teilweise sind sie verschwunden;

Fig. 11 b)

Kompensiertes Multi-Look-Bild. Die bewegten Objekte erscheinen unverzerrt an der geschätzten Referenzposition. Es sind jeweils die Bewegungsrichtung und die geschätzten Geschwindigkeitskomponenten angegeben.

Concrete embodiments of the method according to the invention are explained below, partially with reference to FIGS. 1 to 11. The figures show:

Fig. 1

Generation of a sequence of single-look SAR images a) showing the range migration hyperbola of a static object in the range Doppler plane before the range migration correction (below) and b) after range migration correction

Fig. 2

Measurement of a displacement vector from a pair of images in the sequence (block matching)

Fig. 3

Relationship between the azimuth velocity v _{Mx, a} to be observed in the SL images and the true velocity v _Mx in the azimuth direction without consideration of disturbances

Fig. 4

Observation model for the independent estimation of the azimuth velocity. The noise-disturbing components v _{Mx, m} ^(k) are measured at the output.

Fig. 5

Observation model for the independent estimation of the range speed

Fig. 6

Doppler spectra for one static and one moving object each; The object movement in the direction direction and the resulting Doppler Shift f _M is so large that aliasing occurs in the Doppler spectrum (sampling frequency is the PRF f _p ); a) case a = 1 or 0 (positive range speed); b) case a = -1 or 0 (negative range speed);

Fig. 7

Comparison of the detection and measuring ranges of the object speed for different methods (related to DOSAR parameters, one channel, C-band)

Fig. 8

Estimation results for azimuth and range speed; The standard deviation of the estimation error is shown in each case

Fig. 9

Estimation results for azimuth and range position; The standard deviation of the estimation error is shown in each case;

Fig. 10 a)

Single-look SAR image # 17 from a generated SAR image sequence consisting of 30 images; Look center frequency 75 Hz; The moving objects appear distorted (especially lower object) and away from the street or partially they are completely gone;

Fig. 10b)

Compensated single-look SAR image from the compensated SAR image sequence. The estimated direction of movement of the detected moving objects is indicated by an arrow.

Fig. 11 a)

Multi-look image of a conventional SAR processor. The moving objects appear distorted and shifted in the azimuth direction. Partly they have disappeared;

Fig. 11b)

Compensated multi-look image. The moving objects appear undistorted at the estimated reference position. In each case the direction of movement and the estimated speed components are indicated.

Generation of a SAR image sequence

The SAR image sequence is generated from SAR data of a conventional (ie single-channel) SAR system and consists of temporally consecutive single-look SAR images (SL images). An SL image is an image obtained by using a sub-band from the Doppler spectrum, that is to say a look of a certain look bandwidth, without averaging and therefore with a Lookup bandwidth - maximum azimuth resolution is processed. To generate the sequence of SL images, the range-compressed SAR data are first transmitted in the azimuth direction into the Doppler frequency range by means of a Fourier transformation. The complete Doppler spectrum obtained in this way is subdivided into overlapping bands, each with the same look bandwidth, and an SL image per band is produced (see FIG. 1a). In Fig. 1, the Doppler frequency is denoted by f _D and the PRF by f _p . The images then show the earth's surface from different (horizontal) viewing angles θ ^{( k )} , which in turn depend on the midband frequencies or look center frequencies f ^{( k )} as follows $${\mathrm{\theta }}^{\left(k\right)}=-\arcsin \frac{f^{\left(k\right)}\mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}$$

Where k is the number of the SL image, v _{Ax is} the antenna velocity which is exactly in the azimuth direction x , and λ is the carrier wavelength. Due to the different viewing angles of the individual pictures, one also gets views at different times, and can therefore string the pictures together to form a sequence. The times t ^{( k )} , relative to static objects (index T , English target ) with an azimuth position in the middle of the scene ( x = 0), arise too $$t^{\left(k\right)}\left(R_{T.0}\right)=\frac{R_{T.0}\cdot {\mathrm{tan\theta }}^{\left(k\right)}}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\approx -\frac{R_{T.0}\mathrm{}{f}^{\left(\mathrm{k}\right)}\mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}{2{{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}}^2}\mathrm{With}{f}^{\left(k\right)}=-\frac{2{\mathrm{\upsilon }}_{\mathit{Ax}}\sin {\mathrm{\theta }}^{\left(k\right)}}{\mathrm{\lambda }}$$

Where R _{T , 0 is} the shortest distance between the antenna and the static object. The look bandwidth is adjusted to the speed range to be measured. It must be ensured that it is maximally so large that the fastest objects in the picture are still approximately recognizable as punctiform. In addition, it should at least be so large that the necessary resolution is still achieved.

The images of the sequence undergo ground-range correction and range migration correction. The range migration correction is achieved by shifting the images in the direction y according to the angle of view dependent distance between the antenna and the objects on the earth's surface executed (see Fig. 1 a) / b)). The advantages of this method compared to the conventional [8] consist in an increase in computation time and easier handling in estimating the range speed. This treatment ensures that the static background in the individual images of the sequence appears at the same positions and only the positions of the moving objects change from SL image to SL image.

Detection of candidates for moving objects

• das hellste Pixel des Kandidaten hat eine höhere Intensität als eine von der globalen Intensität des Bildes abhängige Schwelle,
• die Anzahl der zu einem Kandidaten gehörenden Pixel liegt zwischen einem minimalen und einem maximalen Wert; der maximale Wert hängt dabei von der Länge der im Bild sichtbaren Objektform in Azimutrichtung ab und kann demzufolge entsprechend der maximal zu detektierenden Geschwindigkeit gewählt werden,
• die mittlere Intensität der Pixel eines Kandidaten ist merkbar höher als die mittlere Intensität der Nachbarschaft,
• die Pixel eines Kandidaten bilden ein geschlossenes Gebiet,
• kein Pixel des gerade betrachteten Kandidaten berührt ein Pixel eines bereits detektierten anderen Kandidaten.

After the generation of the sequence, a selection of candidates for moving objects takes place in each individual SL image of the sequence. The criterion for the detection as a candidate is an approximately punctiform or ellipsoidal pattern (the latter would occur at an existing azimuth speed). The search for candidates is started at pixels (pixels) with maximum intensity, that is at points with maximum backscatter coefficient (RCS, radar cross section). The start pixel and the surrounding pixels are selected as candidates if the following criteria are met:

• the brightest pixel of the candidate has a higher intensity than a threshold depending on the global intensity of the image,
• the number of pixels belonging to a candidate is between a minimum and a maximum value; the maximum value depends on the length of the object form visible in the image in the azimuth direction and can therefore be selected according to the maximum speed to be detected,
• the average intensity of the pixels of a candidate is noticeably higher than the average intensity of the neighborhood,
• the pixels of a candidate form a closed area,
• no pixel of the currently considered candidate touches a pixel of another candidate already detected.

(Index M für engl. moving target) des Kandidaten im SL-Bild k.After successfully checking these criteria, the candidate is successfully detected in the corresponding SL image. The pixels of the candidate that meet the above conditions form in an advantageous embodiment of the invention a so-called initial form $S_{M.0}^{\left(k\right)}$

(Index M for English moving target ) of the candidate in the SL image k .

After checking these criteria for a start pixel, the same steps are repeated for the remaining start pixels of equal intensity and subsequently for those of lower intensity until an intensity threshold is reached.

aus den verschiedenen SL-Bildern k dem jeweiligen Kandidaten zugeordnet.After evaluating the first SL image k = 1, the remaining SL images are also evaluated so that candidates and corresponding initial shapes are obtained for each SL image. In the next step, the candidate obtained in the individual SL images are associated with each other, that is, an attempt is made a candidate after the initial detection in image k to track k on the basis of the estimated in the next section displacement vectors on the SL-pictures ₀ (engl. Tracking , see next section). Upon successful persecution, the individual initial forms also become $S_{M.0}^{\left(k\right)}$

from the different SL images k associated with the respective candidate.

Speed and position estimation

• Messung der Verschiebungsvektoren (Displacementvektoren von engl. displacement, Verschiebung) zwischen jeweils zwei aufeinanderfolgenden SL-Bildern der Sequenz
• Unabhängige Schätzung der Azimutgeschwindigkeit mit einem Beobachtungsmodell für die Azimutpositionen und einem Maximum-Likelihood-Schätzer - Unabhängige Schätzung der Rangegeschwindigkeit mit einem Beobachtungsmodell für die Rangepositionen und einem Maximum-Likelihood-Schätzer
• Gemeinsame Schätzung von Geschwindigkeit und Position durch ein erweitertes Beobachtungsmodell für Positionen unter Verwendung der ersten Schätzwerte als Startwerte, wobei in einer vorteilhaften Ausführung der Erfindung ein weiterer Parameter zur Kompensation des Aliasing im Dopplerspektrum berücksichtigt wird. In einer vorteilhaften Ausführung der Erfindung werden bei dem Beobachtungsmodell zur gemeinsamen Schätzung von Geschwindigkeit und Position zusätzlich zu den Positionen der Kandidaten in den einzelnen SL-Bildern auch die Amplituden der Kandidaten in den jeweiligen SL-Bildern berücksichtigt.

The speed and position estimation is done for each candidate in several steps:

• Measuring the displacement vectors (Displacementvektoren displacement of engl., Offset) between consecutive two SL-images of the sequence
• Independent estimation of the azimuth velocity with an azimuth position observation model and a maximum likelihood estimator - Independent estimation of the range velocity with an observation model for the range positions and a maximum likelihood estimator
• Common estimation of velocity and position by an extended position observation model using the first estimates as seed values, in an advantageous embodiment of the invention taking into account a further parameter to compensate for aliasing in the Doppler spectrum. In an advantageous embodiment of the invention, in the observation model for the joint estimation of speed and position, in addition to the positions of the candidates in the individual SL images, the amplitudes of the candidates in the respective SL images are also taken into account.

Measurement of the displacement vectors

wobei (x, y) eine der Positionen innerhalb des Fensters in Bild s _k+1 ist, und D _x und D _y die Positionsdifferenzen zwischen den beiden Fenstern bezeichnen. Der Verschiebungsvektor aus dem ersten Bildpaar k ₀ ,k ₀ +1 wird genutzt, um den Kandidaten im Bild k ₀ +2 zu verfolgen, da erwartet wird, dass sich die Bewegungsrichtung des Kandidaten von Bild zu Bild nur unwesentlich verändert. So ist es möglich, den Kandidaten auch über die weiteren Bilder der Sequenz bis zum Bild k ₀ +K _M -1 zu verfolgen, d. h. dem letzten Bild, in dem der Kandidat aufgrund seiner Amplitudenwerte detektiert wurde. K _M ist damit die Gesamtanzahl der Bilder, in denen der Kandidat detektiert wurde.Displacement vectors are measured for a candidate between every two successive images s _k and s _{k + 1} from the sequence. For this purpose, a block matching algorithm can be used, as it is known, for example, from video telephony [9]. In this algorithm, a window is placed at the location of the candidate in image s _{k + 1} so that the candidate is at the center (see Fig. 2). Now in the image s _k, starting from the same position, a second window is moved within a certain search range until the position is reached at which the amplitude values of the pixels in the two windows have the greatest correlation. It is assumed that the positional difference between both windows corresponds to the displacement vector of the candidate between the two images. This displacement vector is assigned to the image s _{k + 1} , so that no displacement vector exists for the first image of the sequence. To measure the correlation different matching criteria can be used, such. Cross correlation coefficient, normalized cross correlation function (NCCF), mean square error (MSE), or mean absolute difference (MAD). The NCCF criterion gives the best results, but in view of a shorter computation time, the MSE criterion can also be used, which yields almost as good results. The NCCF criterion is defined too $$C_{{\mathit{ns}}_{k+1}{s}_k}(D_x.D_y)=\frac{{\displaystyle {\displaystyle \underset{x.y}{\Sigma }}\left[s_{k+1}(x.y)\cdot s_k(x-D_x.y-D_y)\right]}}{\sqrt{{\displaystyle \underset{x.y}{\Sigma }}{s}_{k+1}^2(x.y)\cdot {\displaystyle \underset{x.y}{\Sigma }}{s}_k^2(x-D_x.y-D_y)}}$$

where ( x , y ) is one of the positions within the window in image s _{k + 1} , and D _x and D _y denote the positional differences between the two windows. The displacement vector from the first image pair k ₀ , k ₀ + 1 is used to follow the candidate in the image k ₀ + 2 , since it is expected that the direction of movement of the candidate changes only insignificantly from image to image. It is thus possible to follow the candidate via the further pictures of the sequence as far as the picture k ₀ + K _M - 1 , ie the last picture in which the candidate was detected on the basis of its amplitude values. K _M is thus the total number of images in which the candidate was detected.

Estimation of the azimuth speed

kann die beobachtete Geschwindigkeit in [m/s] unter Berücksichtigung des Bildpunktabstands δx (engl. pixel spacing), der in Azimut- und Rangerichtung als gleich angenommen wird, und der Zeitdifferenz Δt zwischen zwei aufeinanderfolgenden Bildern wie folgt berechnet werden $${\overrightarrow{\mathrm{\upsilon }}}_{M,m}=\frac{{\overrightarrow{D}}^{\left(k\right)}\mathrm{\delta }x}{\mathrm{\Delta }t}$$

From the displacement vector in the unit [pixel / image] ${\overrightarrow{D}}^{\left(k\right)}={(D_x.D_y)}^{\left(k\right)}$

the observed rate can in [m / s] in consideration of the pixel distance δ x (engl. pixel spacing) which is assumed to be equal in azimuth and range direction, and the time difference Δ t is calculated between two successive images as follows $${\overrightarrow{\mathrm{\upsilon }}}_{M.m}=\frac{{\overrightarrow{D}}^{\left(k\right)}\mathrm{\delta }x}{\mathrm{\Delta }t}$$

wobei Δθ die Winkeldifferenz und Δf _k die Differenz der Lookmittenfrequenzen zwischen zwei aufeinanderfolgenden Bildern ist. Die Zeit t ist also nicht überall im Bild gleich, sondern ist von der Range-Entfernung abhängig.The time difference Δ t is obtained by using (2) and the minimum distance R _{M, 0} to between the antenna and the considered candidate moving $$\mathrm{\Delta }t\approx \frac{{\mathrm{\Delta \theta }R}_{M.0}}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\approx -\frac{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M.0}}{2{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}\mathrm{For}\mathrm{sin\theta }\approx \mathrm{\theta }\approx \mathrm{tan\theta }$$

where Δθ is the angular difference and Δ f _{k is} the difference of the look center frequencies between two successive images. The time t is therefore not the same everywhere in the picture, but is dependent on the range distance.

stimmt aus verschiedenen Gründen nicht mit der wirklichen Geschwindigkeit überein:

• in Azimutrichtung durch Stauchung oder Streckung des Spektrums, kombiniert mit Beobachtungszeiten der bewegten Objekte in den SL-Bildern, die von denen der statischen Objekte abweichen
• in Rangerichtung durch die Range-Migration-Korrektur

The observed speed ${\overline{\mathrm{\upsilon }}}_{M.\mathrm{m}}$

does not match the real speed for various reasons:

• in the azimuth direction by compression or extension of the spectrum, combined with observation times of the moving objects in the SL images which deviate from those of the static objects
• in rank direction by the range migration correction

zurückzuführen, mit η als Blickwinkel (entspricht 90° - Einfallswinkel), und ergibt sich zu (vgl. (2)) $${\mathrm{\Delta }t}_M=-\frac{f_M{\mathrm{\lambda }R}_{M,0}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}^2}=\frac{{\mathrm{\upsilon }}_{\mathit{My}}\mathrm{\hspace{1em}sin\eta }\cdot R_{M,0}}{{\mathrm{\upsilon }}_{\mathit{Ax}}^2}$$

In addition, there is a temporal offset of those moving objects that move in the direction direction. This time offset is due to the additional Doppler shift $$f_M=-\frac{2{\mathrm{\upsilon }}_{\mathit{My}}\mathrm{sin\eta }}{\mathrm{\lambda }}$$

with η as the viewing angle (corresponds to 90 ° - angle of incidence), and results (see (2)) $${\mathrm{\Delta }t}_M=-\frac{f_M{\mathrm{\lambda }R}_{M.0}}{2{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}=\frac{{\mathrm{\upsilon }}_{\mathit{My}}\mathrm{sin\eta }\cdot R_{M.0}}{{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}$$

This temporal offset results in the imaging of such objects in other than the expected SL images.

In the following, the two velocity components to be observed and thus the deviation of the observed quantities from the true quantities are first analyzed separately.

In the azimuth direction, the distortions of the velocity can be explained by the fact that the azimuth positions of the moving objects do not coincide with the real positions. Ouchi [10] has shown that the moving objects in the SL images appear at such azimuth positions as if they had twice the speed of reality. However, Ouchi has used, among other things, the approximation v _Mx / v _Ax << 1, which is not satisfied in the case of slow carrier platforms and fast objects. Here, therefore, a more accurate factor m _{v Mx} as the quotient of the analytically derived velocity component to be observed v _{Mx, a} and real velocity component v _Mx derived, which works without this approximation and emanates from the observed azimuth positions p _{Mx of} the moving object: $$m_{{\mathrm{\upsilon }}_{\mathit{Mx}}}=\frac{{\mathrm{\upsilon }}_{\mathit{Mx}.a}}{{\mathrm{\upsilon }}_{\mathit{Mx}}}=\frac{p_{\mathit{Mx}}}{{\mathrm{\upsilon }}_{\mathit{Mx}}t}$$

Since the behavior of the moving objects is almost independent of the azimuth position, one can set the azimuth position to p _{Mx, 0} = p _Mx ( t = 0) = 0 at the time t = 0 without limiting the generality. In (5) then Δ t = t and Δ f _k = f _D , ie the resulting Doppler shift, can be set. Denoting the azimuth position of the antenna with p _Ax , using (1) and (2) $$m_{{\mathrm{\upsilon }}_{\mathit{Mx}}}\approx -\frac{1}{{\mathrm{\upsilon }}_{\mathit{Mx}}}\cdot \frac{p_{\mathit{Ax}}+R_{M.0}\tan \left(\arcsin \frac{{\mathrm{<figure-callout\; id="2"\; label="f\; D\; \lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">f\; </figure-callout>}}_D\mathrm{\lambda }\mathrm{}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}\right)}{\frac{f_D{\mathit{.lambda.r}}_{M.0}}{2{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}}$$

und p _Ax = v _Ax t umformen zu $${m_{{\mathrm{\upsilon }}_{\mathit{Mx}}}}_{\mathrm{\hspace{1em}}}=\frac{{\mathrm{\upsilon }}_{\mathit{Mx},a}}{{\mathrm{\upsilon }}_{\mathit{Mx}}}\approx -\frac{{\mathrm{\upsilon }}_{\mathit{Ax}}}{{\mathrm{\upsilon }}_{\mathit{Mx}}}\cdot (1-\frac{{\mathrm{\upsilon }}_{\mathit{Ax}}^2}{{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}^2})$$

If one again makes the approximation sinθ ≈ θ ≈ tanθ, one can do this with $$f_D=-\frac{2({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}{\mathrm{\lambda }}\cdot \mathrm{sin\theta }$$

and p _Ax = v _Ax t to form $${m_{{\mathrm{\upsilon }}_{\mathit{Mx}}}}_{\mathrm{}}=\frac{{\mathrm{\upsilon }}_{\mathit{Mx}.a}}{{\mathrm{\upsilon }}_{\mathit{Mx}}}\approx -\frac{{\mathrm{\upsilon }}_{\mathit{Ax}}}{{\mathrm{\upsilon }}_{\mathit{Mx}}}\cdot (1-\frac{{\mathrm{\upsilon }}_{\mathit{Ax}}^2}{{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}^2})$$

Hence, for the underlying real speed too $${\mathrm{\upsilon }}_{\mathit{Mx}}=f^{-1}\left({\mathrm{\upsilon }}_{\mathit{Mx}.a}\right)={\mathrm{\upsilon }}_{\mathit{Ax}}-\sqrt{\frac{{\mathrm{<figure-callout\; id="3"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}{{\mathrm{\upsilon }}_{\mathit{Mx}.a}+{\mathrm{\upsilon }}_{\mathit{Ax}}}}{\mathrm{for\; \upsilon }}_{\mathit{Mx}.a}>-{\mathrm{\upsilon }}_{\mathit{Ax}}$$

zu $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx},1}^{\left(k\right)}={\mathrm{\upsilon }}_{\mathit{Ax}}\cdot [1-\frac{1}{\sqrt{1-\frac{2D_x^{\left(k\right)}{\mathrm{\delta }x\mathrm{\upsilon }}_{\mathit{Ax}}}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M,0}}}}]{\mathrm{\hspace{1em}für\hspace{1em}\upsilon }}_{\mathit{Mx},a}>-{\mathrm{\upsilon }}_{\mathit{Ax}}$$

This relationship is shown graphically in FIG. 3 for v _Ax = 82.5 m / s = 297 km / h. It can be seen that this relation for small velocities v _Mx agrees with the factor 2 found by Ouchi. Assuming that the displacement vectors reflect the real displacements of the candidate from image k -1 to image k, ie are undisturbed, the inverse of Eq: (12) represents a relationship between the true velocity v _Mx and that measured by the displacement vectors Speed. With Eq. (4), therefore, (12) yields a first estimate of the velocity v _Mx in the image k as a function of the displacement vector component $D_x^{\left(k\right)}$

to $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx}.1}^{\left(k\right)}={\mathrm{\upsilon }}_{\mathit{Ax}}\cdot [1-\frac{1}{\sqrt{1-\frac{2D_x^{\left(k\right)}{\mathrm{\delta }x\mathrm{\upsilon }}_{\mathit{Ax}}}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M.0}}}}]{\mathrm{for\; \upsilon }}_{\mathit{Mx}.a}>-{\mathrm{\upsilon }}_{\mathit{Ax}}$$

am wahrscheinlichsten hervorgegangen sind: $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx},\mathit{ML}}=\underset{{\mathrm{\upsilon }}_{\mathit{Mx}}}{\mathrm{arg\hspace{1em}max}}\left(p_{{\mathrm{\upsilon }}_{\mathit{Mx},m}|{\mathrm{\upsilon }}_{\mathit{Mx}}}({{\mathrm{\upsilon }}_{\mathit{Mx},m}}^{\left(k_0\right)},\mathrm{...},{{\mathrm{\upsilon }}_{\mathit{Mx},m}}^{(k_0+k_M-1)}|{\mathrm{\upsilon }}_{\mathit{Mx}})\right)$$

Usually, however, the observed variables, ie the displacement vectors in this case, are not undisturbed. In an observation model, it can be assumed that all disturbances taken together are represented by an additive normally distributed noise n ^(k) (see FIG. 4). Starting from this observation model, the so-called maximum likelihood estimation method is suitable for obtaining an optimal estimate. In this estimation method, no a priori knowledge of the probability density of the size to be estimated is necessary [11]. With the maximum likelihood estimation rule one obtains that value v _Mx as the estimated value ν _{Mx, ML} , from which the measured values ${\mathrm{\upsilon }}_{\mathit{Mx}.m}^{\left(k\right)}$

most likely to have emerged: $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx}.\mathit{ML}}=\underset{{\mathrm{\upsilon }}_{\mathit{Mx}}}{\mathrm{arg\; max}}\left(p_{{\mathrm{\upsilon }}_{\mathit{Mx}.m}|{\mathrm{\upsilon }}_{\mathit{Mx}}}({{\mathrm{\upsilon }}_{\mathit{Mx}.m}}^{\left(k_0\right)}.\mathrm{...}.{{\mathrm{\upsilon }}_{\mathit{Mx}.m}}^{({\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+k_M-1)}|{\mathrm{\upsilon }}_{\mathit{Mx}})\right)$$

Assuming statistical independence of the disturbances in adjacent images, the conditional compound probability density in Eq. (14) as a product of normal distributions. After the onset of this composite probability density, Eq. (14) derived to determine the estimated value and set to zero. More simply, the maximum likelihood estimate can be determined by deriving the negative logarithm of the compound probability density, the so-called ML cost function. With the context according to Eq. (12) results in an estimate of the azimuth velocity $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx}.\mathit{ML}}=f^{-1}\left[\frac{1}{K_M}\underset{k=k_0}{\overset{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+K_M-1}{\Sigma }}{\mathrm{\upsilon }}_{\mathit{Mx}.\mathit{m}}^{\left(\mathit{k}\right)}\right]\mathrm{,}$$

zu $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx},\mathit{ML}}=f^{-1}\left[\frac{1}{K_M}\sum _{k=k_0}^{k_0+K_M-1}\frac{2D_x^{\left(k\right)}{{\mathrm{\delta }x\mathrm{\upsilon }}_{\mathit{Ax}}}^2}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M,0}}\right]\mathrm{.}$$

After inserting Eq. (4) and (2), the maximum likelihood estimate (ML) is finally obtained as a function of the displacement vector components $D_x^{\left(k\right)}$

to $${\hat{\mathrm{\upsilon }}}_{\mathit{Mx}.\mathit{ML}}=f^{-1}\left[\frac{1}{K_M}\underset{k=k_0}{\overset{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+K_M-1}{\Sigma }}\frac{2D_x^{\left(k\right)}{{\mathrm{\delta }\mathrm{<figure-callout\; id="2"\; label="x\; \upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">x\; </figure-callout>}\mathrm{\upsilon }}_{\mathit{Ax}}}^2}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M.0}}\right]\mathrm{,}$$

Estimation of the range speed

zwischen jeweils zwei aufeinanderfolgenden SL-Bildern korrigiert werden. Einen Maximum-Likelihood-Schätzwert für die Rangegeschwindigkeit erhält man damit analog zur Azimutgeschwindigkeit mit dem Beobachtungsmodell gemäß Fig. 5 zu $${\hat{\mathrm{\upsilon }}}_{\mathit{My},\mathit{ML}}=\frac{2{\mathrm{\upsilon }}_{\mathit{Ax}}^2\cdot \mathrm{\delta }y}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M,0}{K}_M}\left[(-\sum _{k=k_0}^{k_0+K_M-1}{D}_{\mathit{y}}^{\left(k\right)})+\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{({\mathit{k}}_0-1)}\left({\mathit{R}}_{\mathit{M},0}\right)-\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{({\mathit{k}}_0+{\mathit{K}}_{\mathit{M}}-1)}\left({\mathit{R}}_{\mathit{M},0}\right)\right],$$

wobei der Pixelabstand in Rangerichtung mit δy bezeichnet ist. Die darin enthaltenen Terme aus der Range-Migration-Korrektur ergeben sich zu $$\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{\left(\mathit{k}\right)}\left(R_{M,0}\right)=(\sqrt{{\left({\mathit{t}}^{\left(k\right)}\left(R_{M,0}\right)\cdot {\mathrm{\upsilon }}_{\mathit{Ax}}\right)}^2+R_{M,0}^2}-R_{M,0})\cdot \frac{\mathrm{sin\eta }}{\mathrm{\delta }y}$$

As already mentioned, the velocity component obtained from the displacement vectors in direction direction v _{My, m is} also corrupted, but in a different way than the azimuth component. In the range speed, the corruption is due to the fact that the range positions of those objects whose range speed is not equal to zero are offset by the range migration correction. Nevertheless, a first estimate for the range speed can be obtained with the aid of the displacement vectors. The range components of the displacement vectors must be around the position differences caused by the range migration correction $\mathrm{\Delta }{y}_T^{\left(k\right)}$

be corrected between each two successive SL images. A maximum likelihood estimate for the range speed is thus obtained analogously to the azimuth speed with the observation model according to FIG. 5 $${\hat{\mathrm{\upsilon }}}_{\mathit{My}.\mathit{ML}}=\frac{2{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}\cdot \mathrm{\delta }y}{{\mathrm{\Delta }f}_k{\mathrm{\lambda }R}_{M.0}{K}_M}\left[(-\underset{k=k_0}{\overset{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+K_M-1}{\Sigma }}{D}_{\mathit{y}}^{\left(k\right)})+\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{({\mathit{k}}_0-1)}\left({\mathit{R}}_{\mathit{M}.0}\right)-\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{({\mathit{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+{\mathit{K}}_{\mathit{M}}-1)}\left({\mathit{R}}_{\mathit{M}.0}\right)\right].$$

wherein the pixel pitch in the direction direction is denoted by δy . The terms contained in the range migration correction result from this $$\mathrm{\Delta }{\mathit{y}}_{\mathit{T}}^{\left(\mathit{k}\right)}\left(R_{M.0}\right)=(\sqrt{{\left({\mathit{t}}^{\left(k\right)}\left(R_{M.0}\right)\cdot {\mathrm{\upsilon }}_{\mathit{Ax}}\right)}^2+R_{M.0}^2}-R_{M.0})\cdot \frac{\mathrm{sin\eta }}{\mathrm{\delta }y}$$

Where t ^{( k )} ( R ) is the time from Eq. (2) and η the viewing angle (corresponds to 90 ° - angle of incidence).

The estimated value according to Eq. (17) is relatively inaccurate because the position shifts from the range migration correction are almost as large as those from the object motion itself. More accurate estimation of this component is therefore made along with the azimuth component and the position in a second step. In an advantageous embodiment of the invention, it is possible to further increase the accuracy by including the amplitude values in the individual SL images.

Joint speed and position estimation

The common speed and position estimate is a significant innovation compared to DE 196 18 155 C2. It allows the detection of the movement of the objects almost independent and the magnification of the measuring ranges for the two speed components. In addition, hereby a - for single-channel SAR systems - very accurate position estimation is realized, which is also not included in DE 196 18 155 C2.

aus den SL-Bildern als Startwerte für die Position. Die Azimutkomponente der Startposition wird dabei um den von der Rangegeschwindigkeit verursachten Versatz $${\mathrm{\Delta }x}_M=-\frac{{\mathrm{\upsilon }}_{\mathit{My}}\mathrm{\hspace{1em}sin\eta }}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot R_{M,0}$$

korrigiert, wobei für ν _My an dieser Stelle der Schätzwert ν̂ _My,ML aus Gl. (17) verwendet wird.As described in the first section, the essential effect of the object velocity in the direction of the direction is an offset in the azimuth direction. It is thus given a double meaning of an offset due to the range speed moving object with a static object at a deviating by this offset azimuth position. Here, therefore, a method is explained how this ambiguity can be corrected by evaluating the positions in the individual images of the sequence or the estimate of speed and position at Evaluation of this information can be improved. Advantageously, in addition to the evaluation of the positions, the time-dependent signal amplitude can also be taken into account. For this purpose, the range speed and the azimuth position are jointly estimated according to the invention. In addition to the dependence of the measured azimuth position on the range velocity, there is also a slight dependence of the measured displacement vectors D ^{( k )} on the position, so that four parameters p _{Mx , 0} , p _{My , 0} , ν _Mx and ν _My as well as an additionally introduced parameter a for aliasing compensation should be estimated jointly by means of a common observation model and an optimization method. Input values for the optimization method are the maximum likelihood estimates from the previous section for the velocity (Eqs. (16) and (17)) and the averaged individual positions ${\overrightarrow{p}}_M^{\left(k\right)}$

from the SL images as starting values for the position. The azimuth component of the start position is thereby the offset caused by the range speed $${\mathrm{\Delta }x}_M=-\frac{{\mathrm{\upsilon }}_{\mathit{My}}\mathrm{sin\eta }}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot R_{M.0}$$

corrected, where for ν _My at this point the estimated value ν _{My, ML} from Eq. (17) is used.

In order to be able to carry out the joint velocity and position estimation, analytical relationships for azimuth and range components of the time-dependent, error-prone position of an arbitrarily moved object are derived. Advantageously, this analytical relationship is extended by the time-dependent amplitude of the object. In this third relationship enters the range speed and the attenuation of the signal through the antenna diagram. In addition, the effects of aliasing in the Doppler spectrum are considered in the analytic relationships. This aliasing is caused by a large range velocity, which is the spectral components of such a moving object according to the Doppler shift according to Eq. (5) shifts beyond half the sampling frequency f _p (see FIG. 6). This mimics such objects in other images of the sequence (e.g. at the end of the sequence instead of at the beginning) and also the change of the amplitude is additionally shifted in time. By means of the derived analytic relations for the time-dependent positions, a cost function is set up, which deals with the quadratic errors between the measured and the analytically derived positions. Advantageously, the cost function is extended by the derived analytical relationship for the time-dependent amplitude by taking into account as an additional cost term the quadratic errors between measured and analytically calculated amplitudes. Of course, higher-order errors and / or absolute errors in the cost function can also be taken into account. The aim of the subsequent optimization procedure is to minimize the costs by varying the five parameters, since in a cost function low costs correspond with a high probability of the parameter combination.

wobei f _k der Dopplerfrequenz des bewegten Objekts entsprechen muß. R' _M ist die durch die Azimutverschiebung verfälschte, zeitveränderliche Rangeentfernung. Mit $t_M^{\left(k\right)}$

ist jeweils der Zeitpunkt bezeichnet, zu dem das bewegte Objekt im SL-Bild k abgebildet wird. Infolge der Rangegeschwindigkeit v _My können Dopplerfrequenzen auftreten, die im Basisband des Dopplerspektrums (a = 0) oder bei Auftreten von Aliasing außerhalb (a = -1 oder a = 1) liegen (siehe Fig. 6). Da der Parameter a für das gleiche Objekt über den Verlauf der Sequenz von SL-Bildern zwei unterschiedliche Werte annehmen kann, wird er mit dem hochgestellten Index (k) versehen. Allerdings bleibt in aufeinanderfolgenden Bildern der Sequenz, in denen ein Kandidat detektiert wurde, der Parameter a ^(k) konstant. Damit wird Gl. (20) zu $${X^{\prime }}_{M,a}^{\left(k\right)}={\mathrm{\upsilon }}_{\mathit{Ax}}{t}_M^{\left(k\right)}+{R^{\prime }}_M\left(t_M^{\left(k\right)}\right)\frac{(f_D+a^{\left(k\right)}\cdot f_p)\cdot \mathrm{\lambda }}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}$$

The time-dependent, error-prone azimuth position of an object moving in range and azimuth direction at the reference position p _{Mx, 0} = p _Mx ( t = 0) = 0 is obtained analytically with Eq. (1) too $$\begin{array}{c}{X^{\text{'}}}_{M.a}^{\left(k\right)}=p_{\mathit{Ax}}\left(t_M^{\left(k\right)}\right)-{R^{\text{'}}}_M\mathrm{sin\theta }\left(t_M^{\left(k\right)}\right)\\ {X^{\text{'}}}_{M.a}^{\left(k\right)}={\mathrm{\upsilon }}_{\mathit{Ax}}{\mathrm{t}}_M^{\left(k\right)}-{R^{\text{'}}}_M\left(t_M^{\left(k\right)}\right)\frac{{\mathrm{<figure-callout\; id="2"\; label="f\; k\; \lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">f\; </figure-callout>}}_k\mathrm{\lambda }\mathrm{}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}\\ \mathrm{}\end{array}$$

where f _k must correspond to the Doppler frequency of the moving object. R ' _M is the time-varying range distance distorted by the azimuth shift. With $t_M^{\left(k\right)}$

in each case the time is designated at which the moving object is imaged in the SL image k . As a result of the range velocity v _My , Doppler frequencies can occur which lie in the baseband of the Doppler spectrum ( a = 0) or when aliasing occurs outside ( a = -1 or a = 1) (see FIG. 6). Since the parameter a for the same object can assume two different values over the course of the sequence of SL images, it is provided with the superscript (k). However, in successive frames of the sequence in which a candidate was detected, the parameter a ^(k) remains constant. Thus Eq. (20) too $${X^{\text{'}}}_{M.a}^{\left(k\right)}={\mathrm{\upsilon }}_{\mathit{Ax}}{t}_M^{\left(k\right)}+{R^{\text{'}}}_M\left(t_M^{\left(k\right)}\right)\frac{(f_D+a^{\left(k\right)}\cdot f_p)\cdot \mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}$$

ergibt sich schließlich mit Gl. (11) $$\begin{array}{c}{X^{\prime }}_{M,a}^{\left(k\right)}\approx ({\mathrm{\upsilon }}_{\mathit{Ax}}-\frac{{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}^2}{{\mathrm{\upsilon }}_{\mathit{Ax}}})t_M^{\left(k\right)}+\frac{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot p_{\mathit{Mx},0}+\\ +{R^{\prime }}_M\left(t_M^{\left(k\right)}\right)(\frac{a^{\left(k\right)}\cdot f_p\cdot \mathrm{\lambda }}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}-\frac{{\mathrm{\upsilon }}_{\mathit{My}}}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot {\mathrm{sin\eta cos\theta }}_{\mathit{k}})\end{array}$$

After the onset of the relationship $$f_D\left(t\right)=-\frac{2({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}{\mathrm{\lambda }}\cdot \mathrm{sin\theta }-\frac{2{\mathrm{\upsilon }}_{\mathit{My}}}{\mathrm{\lambda }}\cdot \mathrm{sin\eta cos\theta }$$

finally results with Eq. (11) $$\begin{array}{c}{X^{\text{'}}}_{M.a}^{\left(k\right)}\approx ({\mathrm{\upsilon }}_{\mathit{Ax}}-\frac{{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}^2}{{\mathrm{\upsilon }}_{\mathit{Ax}}})t_M^{\left(k\right)}+\frac{({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}})}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot p_{\mathit{Mx}.0}+\\ +{R^{\text{'}}}_M\left(t_M^{\left(k\right)}\right)(\frac{a^{\left(k\right)}\cdot f_p\cdot \mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}}-\frac{{\mathrm{\upsilon }}_{\mathit{My}}}{{\mathrm{\upsilon }}_{\mathit{Ax}}}\cdot {\mathrm{sin\eta cos\theta }}_{\mathit{k}})\end{array}$$

One recognizes here a dependence on p _{Mx, 0} , ν _Mx , ν _My and a ^{( k )} .

ergeben sich analytisch mit Gl. (18) zu $${Y^{\prime }}_{M,a}^{\left(k\right)}=\sqrt{{\left[({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}}){\mathrm{t}}_M^{\left(k\right)}-p_{\mathit{Mx},0}\right]}^2+{[{\mathrm{\upsilon }}_{\mathit{My}}{t}_M^{\left(k\right)}+p_{\mathit{My},0}]}^2}+\mathrm{\Delta }{y}_T^{\left(k\right)},$$

d. h. sie sind von p _Mx,0, p _My,0, ν _Mx und ν _My abhängig. Schließlich kann vorteilhaft ein analytischer Wert für die Signalamplitude im Bild k unter Vernachlässigung von Störungen wie z. B. richtungsabhängiger Reflexionen zu $$\left|{u_a}^{\left(k\right)}\right|=\left|u_{\mathit{max}}\right|\cdot {\mathit{si}}^2(\mathrm{\pi }\frac{l}{\mathrm{\lambda }}\cdot {\mathrm{sin\theta }}_k)$$

angegeben werden, wobei u _max der maximal auftretende Amplitudenwert des jeweiligen Objekts ist und Sättigungseffekte unberücksichtigt bleiben. Im Fall des Auftretens von Aliasing verfälscht sich auch der zeitliche Versatz nach Gl. (7) zu $${{\mathrm{\Delta }t}_M}^{\left(k\right)}=-\frac{(f_D+a^{\left(k\right)}\cdot f_p){\mathrm{\lambda }R}_{M,0}}{2{\mathrm{\upsilon }}_{\mathit{Ax}}^2},$$

so dass dann auch die Amplituden in den SL-Bildern nach Gl. (25) entsprechend zeitlich verschoben erscheinen.The ground range positions ${Y^{\text{'}}}_M^{\left(k\right)}$

result analytically with Eq. (18) too $${Y^{\text{'}}}_{M.a}^{\left(k\right)}=\sqrt{{\left[({\mathrm{\upsilon }}_{\mathit{Ax}}-{\mathrm{\upsilon }}_{\mathit{Mx}}){\mathrm{t}}_M^{\left(k\right)}-p_{\mathit{Mx}.0}\right]}^2+{[{\mathrm{\upsilon }}_{\mathit{My}}{t}_M^{\left(k\right)}+p_{\mathit{My}.0}]}^2}+\mathrm{\Delta }{y}_T^{\left(k\right)}.$$

ie they depend on p _{Mx , 0} , p _{My , 0} , ν _Mx and ν _My . Finally, advantageously, an analytical value for the signal amplitude in the image k neglecting disturbances such. B. directional reflections $$\left|{u_a}^{\left(k\right)}\right|=\left|u_{\mathit{Max}}\right|\cdot {\mathit{si}}^2(\mathrm{\pi }\frac{l}{\mathrm{\lambda }}\cdot {\mathrm{sin\theta }}_k)$$

where u _{max is} the maximum occurring amplitude value of the respective object and saturation effects are disregarded. In the case of the occurrence of aliasing, the time offset according to Eq. (7) too $${{\mathrm{\Delta }t}_M}^{\left(k\right)}=-\frac{(f_D+a^{\left(k\right)}\cdot f_p){\mathrm{\lambda }R}_{M.0}}{2{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; Ax"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_{\mathit{Ax}}^{\mathit{}}}.$$

so that then the amplitudes in the SL-images according to Eq. (25) appear to be postponed accordingly.

ergibt.For the joint estimation, the quadratic error between the measured and the analytically derived positions according to Eq. (23) and (24) and inserted in the cost function already mentioned $$\mathrm{\zeta }=\frac{1}{K_M}\underset{k_0}{\overset{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+K_M-1}{\Sigma }}{({X^{\text{'}}}_M^{\left(k\right)}-{\widehat{X^{\text{'}}}}_M^{\left(k\right)})}^2+{({Y^{\text{'}}}_M^{\left(k\right)}-{\widehat{Y^{\text{'}}}}_M^{\left(k\right)})}^2$$

results.

In an advantageous embodiment of the invention, in the cost function according to Eq. (27) additionally the measured and the according to Eq. (25) accounted for analytically derived amplitudes. The extended cost function thus results $$\mathrm{\zeta }=\frac{1}{K_M}\underset{k_0}{\overset{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+K_M-1}{\Sigma }}{({X^{\text{'}}}_M^{\left(k\right)}-{\widehat{X^{\text{'}}}}_M^{\left(k\right)})}^2+{({Y^{\text{'}}}_M^{\left(k\right)}-{\widehat{Y^{\text{'}}}}_M^{\left(k\right)})}^2+{\mathrm{\lambda }}_A\cdot {(\frac{u^{\left(k\right)}}{\overline{u}}-\frac{{\hat{u}}^{\left(k\right)}}{\overline{\hat{u}}})}^2$$

Since in Eq. (27) and Eq. (28) into the analytically calculated values according to Eq. (23) to (25), respectively, the values to be estimated in the parameters, beginning with the values from the preceding sections, have been replaced by the a. The measurements of the amplitude u ^(k) and the position (X _'m, Y' _m) ^(k) are made in the SL-images. For this purpose, the center of gravity of an observed object is used in each case. In order to achieve independence from the backscatter coefficient, ie from u _max , the amplitude term in each case is divided by the mean amplitude u or û . The averaging is done over all values u ^{( k )} and û ^{( k )} , in which the candidate was detected. In addition, the quadratic errors of amplitude are weighted with a parameter λ _A to establish the relation of quadratic errors.
In an exemplary embodiment of the method according to the invention, the established cost function for obtaining the improved estimates for the five parameters p _{Mx , 0} , p _{My , 0} , ν _Mx , ν _My and a is minimized, for example, using the downhill simplex optimization method [12]. , Starting values for the optimization are, as mentioned, the previously obtained estimates from the preceding sections. The downhill simplex optimization method changes one of the parameters as long as the costs decrease according to the given cost function or worsen in a maximum of N _{op t, worse} iterations. Then the next parameter is changed, etc. During the search, the step size is reduced. Canceling after a fixed number of iterations or if no change in the parameter results in a reduction of the cost after more than N _{Op t, worse} steps. The downhill simplex optimization method is convergent and can overcome local minima.

In an advantageous embodiment of the invention, an estimated value ν _Mx for the azimuth component of the velocity vector of the candidate is determined from the azimuth components of the displacement vectors D _x ^{(k) of} the candidate. This estimate is then kept constant in the optimization method for estimating the other parameters.

des Kandidaten in den einzelnen single-look SAR-Bildern ein Schätzwert für die Rangekomponente p̂ _My,0 des Positionsvektors des Kandidaten ermittelt, der bei der Optimierungsmethode zur Schätzung der übrigen zu schätzenden Parameter konstant gehalten wird.In a further advantageous embodiment of the invention is from the individual range positions ${Y^{\text{'}}}_M^{\left(k\right)}$

the candidate in each single-look SAR images, an estimated value for the range component p _{My , 0 of} the position vector of the candidate is determined, which is kept constant in the optimization method for estimating the other parameters to be estimated.

Verification of candidates as moving objects

• die Verfolgung eines Kandidaten von SL-Bild k ₀ zu SL-Bild k ₀ +1 und weiter bis SL-Bild k ₀ +K _M -1 mit Hilfe der Verschiebungsvektoren über beispielsweise mindestens K _M,min = 4 SL-Bilder der Sequenz möglich ist. Dabei ist eine Unterbrechung der Verfolgung in ein oder zwei aufeinanderfolgenden SL-Bildern zulässig. Außerdem muß erfüllt sein, dass
• einer der beiden Beträge der Geschwindigkeitskomponenten folgende Mindestwerte erreicht:
  Azimutkomponente |ν̂ _Mx |: 5 km/h oder Rangekomponente |ν̂ _My |: 20 km/h

As already explained in the section 'Speed and position estimation', the candidates are tracked from image to image according to the displacement vectors estimated with previous image pairs. The selected candidates are verified as moving objects when

• tracking a candidate from SL image k ₀ to SL image k ₀ +1 and further to SL image k ₀ + K _M - 1 using the displacement vectors over, for example, at least K _{M, min} = 4 SL images of the sequence is. It is permissible to interrupt the tracking in one or two consecutive SL-pictures. It must also be satisfied that
• one of the two amounts of speed components has the following minimum values:
  Azimuth component | ν _Mx |: 5 km / h or range component | ν _My |: 20 km / h

Compensation of aberrations

The compensation of the aberrations with the aim of the correct representation of the moving objects can be done in two ways by generating a compensated SAR image sequence or a compensated multi-look image. The background is preserved in both cases.

• Extraktion der bewegten Objekte mit den Initialformen $S_{M,0}^{\left(k\right)}$
  
  aus den ursprünglichen SL-Bildern
• Berechnung der Positionen (X _M , Y _M ) ^(k) in den einzelnen SL-Bildern anhand der gewonnenen Schätzwerte für Position und Geschwindigkeit und den Zeiten der Bilder gemäß Gl. (2)
• Einsetzen der bewegten Objekte an den berechneten Positionen (X _M , Y _M ) ^(k) mit der Form S _M.. Der Zeitversatz nach Gl. (26) wird dadurch kompensiert, dass die Objekte in entsprechend anderen Bildern extrahiert werden als sie wieder eingesetzt werden.

In the first case of the compensated SAR image sequence, the following steps are performed in the individual SL images:

• Extraction of the moving objects with the initial forms $S_{M.0}^{\left(k\right)}$
  
  from the original SL pictures
• Calculation of the positions (X _M , Y _M ) ^(k) in the individual SL images on the basis of the obtained position and velocity estimates and the times of the images according to Eq. (2)
• Inserting the moving objects at the calculated positions (X _M , Y _M ) ^{(k) of} the form S _M. , The time offset according to Eq. (26) is compensated by extracting the objects according to different images than they are used again.

An exemplary result can be seen in FIG. 10 b). In comparison to the original image in FIG. 10 a), all aberrations are largely compensated.

• Extraktion der bewegten Objekte mit den Initialformen $S_{M,0}^{\left(k\right)}$
  
  aus den ursprünglichen SL-Bildern
• Einsetzen der bewegten Objekte an der geschätzten Referenzposition p̂ _M,0 mit der Form S _M . Auch hier wird der Zeitversatz nach Gl. (26) durch Extrahieren und Einsetzen in u. U. unterschiedlichen Bildern kompensiert.
• Multi-Look-Mittelung, d. h. Mittelung der Amplitudenwerte aus verschiedenen SL-Bildern

In the second case of the compensated multi-look image, the following steps are performed in the individual SL images:

• Extraction of the moving objects with the initial forms $S_{M.0}^{\left(k\right)}$
  
  from the original SL pictures
• Inserting the moving objects at the estimated reference position p _{M , 0} with the form S _M. Again, the time offset according to Eq. (26) by extracting and inserting in u. U. different images compensated.
• Multi-look averaging, ie averaging of the amplitude values from different SL images

An exemplary result can be seen in FIG. 11 b). In comparison, Fig. 11 a) shows a multi-look image of a conventional SAR processor. It can be clearly seen the aberrations of the moving objects, which are largely compensated in Fig. 11 b).

Results

The inventive method has been applied to simulated and true SAR raw data. The simulated data was simulated raw SAR data from moving objects overlaid with real background data (clutter data). Both the real SAR raw data of moving vehicles and the background data used for superimposing come from the DOSAR system of the company Dornier GmbH in C-band (5.3 GHz) or X-band (9.6 GHz). The signal-to-clutter ratio in the SL image was approximately between 3 and 8 dB. The antenna speed was about 300 km / h or 83 m / s, and a sequence of 30 images was generated and used. The thereby utilized Doppler tape width was 1500 Hz, which corresponds to about 2.5 times the Clutterbandbreite.

The azimuth velocity v _Mx could be measured with azimuth speeds between +/- 5 km / h (+/- 1.4 m / s) and +/- 120 km / h (+/- 33.3 m / s) with one standard deviation estimation error of 2.7 km / h. For the range speed the ensprechende value was 8.5 km / h for the range between +/- 20 km / h (+/- 5.6 m / s) and +/- 120 km / h (+/- 33.3 m / s) (see Fig. 8). Compared to The other methods known from the literature have extended the measuring range for both speed components (see Fig. 7), with some ranges (for low speeds and the range | v _Mx | <22 km / h and 80 km / h <| v _My | < 120 km / h) for the first time by a method for single-channel SAR data. The upper value of the range speed measurement range was determined for the 1500 Hz PRF available at the DOSAR system at a carrier frequency in the C band (5.3 GHz).

The detection rate is as follows: In the range between +/- 10 km / h (+/- 2.8 m / s) and +/- 80 km / h (+/- 22.2 m / s), 96% of all moving objects detected, at a higher or lower speed at least 80%. The false alarm rate was about 2%. Detection and false alarm rates can be easily controlled by the minimum number K _{M, min of} the images in which a candidate must be detected and the speed thresholds. For the given results, the minimum number was K _{M, min} = 4 images.

In the position estimation, the following results were obtained (see Fig. 9): The standard deviation of the estimation error is 28.5 m in azimuth and 5.7 m in direction direction. By comparison, in [6] about 50 m are given for both components. Thus, overall, both for the estimated speed and for the position results in a significant improvement over previously known methods.

This is also reflected in the results obtained. 10 shows an exemplary SL image from a SAR image sequence (FIG. 10 a, detected moving objects are marked with a white square) in comparison to a compensated SL image (b). While in the original image, the moving objects are distorted and appear off the road or have partially disappeared completely, these aberrations in Fig. 10 b) are largely compensated. The same is true in the case of multi-look images. FIG. 11 a) shows a multi-look image of a conventional SAR processor, in which the distortions of the moving objects emerge particularly clearly. FIG. 11 b) shows in comparison a compensated multi-look image in which the objects are again imaged largely undistorted on the road.

literature

• [1] R.K. Raney, "Synthetic aperture imaging radar and moving targets", IEEE Trans. Aerospace and Electronic Systems, Vol. AES-7, No. 3, pp. 499-505, May 1971.
• [2] Y. Wang, R. Chellappa and Q. Zheng, "CFAR detection of targets in fully polarimetric SAR images", Technical Report, Department of Computer Science, University of Maryland, College Park, Nov. 1993.
• [3] A. Freeman and A. Currie, "Synthetic aperture radar (SAR) images of moving targets", GEC Journal of Research, Vol. 5, No. 2,1987.
• [4] E. D'Addio, M. Di Bisceglie and S. Bottalico, "Detection of moving objects with airborne SAR", Signal Processing, Vol. 36, No. 2, pp. 149-162, March 1994.
• [5] S. Barbarossa and A. Farina: "Detection and imaging of moving objects with synthetic aperture radar, Part 2: Joint time-frequency analysis by Wigner-Ville distribution." In IEE Proceedings-F, Vol. 139, No. 1, pp. 89-97, Feb. 1992.
• [6] J.R. Moreira and W. Keydel, "A new MTI-SAR approach using the reflectivity displacement method." IEEE Trans. Geoscience and Remote Sensing, Vol. 5, pp. 1238-1244, Sept. 1995.
• [7] M. Kirscht, "Detection, velocity estimation and imaging of moving targets with single-channel SAR." In Proc. of European Conference on Synthetic Aperture Radar, EUSAR'98, Friedrichshafen, Germany, pp. 587-590, May 1998.
• [8] J.C. Curlander and R.N. McDonough: Synthetic Aperture Radar: Systems and Signal Processing, New York: John Wiley & Sons, 1991, pp. 193f.
• [9] M. Bierling, "Displacement estimation by hierarchical block matching", 3rd SPIE Symposium on Visual Communications and Image Processing, Cambridge, USA, pp. 942-951, 1988.
• [10] K. Ouchi, "On the multilook images of moving targets by synthetic aperture radars", IEEE Trans. Antennas and Propagation, Vol. AP-33, No. 8, pp. 823-827, 1985.
• [11] J.L. Melsa and D.L. Cohn, Decision and Estimation Theory, Tokyo, Japan: McGraw-Hill, 1978.
• [12] J.A. Nelder and R. Mead, A Simplex Method for Function Minimization, Computer Journal, No. 7, pp. 308-317, July 1964.

Claims (8)

1. Method for detection as well as speed and position estimation of moving objects in SAR images, comprising the following method steps:

   - generation of a sequence of successive single-look SAR images with the same polarization and of a look mid-frequency, which varies from one image to the next;

   - detection of candidates for moving objects in the single-look SAR images in the sequence by searching for regions with an intensity profile which differs from that of the surrounding area;

   - speed estimation of the detected candidates by estimation of one or more shift vectors for each candidate using the correlation between successive single-look SAR images;

   - verification of the detected candidates as moving objects,

   
   characterized in that the estimation of one or more speed components of a detected candidate is carried out together with the estimation of one or more position components of the candidate, with a cost function which is dependent on the components to be estimated being set up taking into the account the positions of the candidate in the individual single-look SAR images in the sequence, and being minimized by means of an optimization method for the parameter or parameters to be estimated.
2. Method according to Claim 1, characterized in that the amplitudes of the candidates in the individual single-look SAR images in the sequence are also taken into account in order to set up the cost function.
3. Method according to Claim 1 or 2, characterized in that a further parameter to be estimated is taken into account, in order to compensate for the aliasing in the Doppler spectrum while minimizing the cost function.
4. Method according to one of the preceding claims, characterized in that an estimated value for the azimuth component of the speed vector of the respective candidate is determined from the azimuth components of the shift vectors of a candidate, and is kept constant during the optimization method for joint position and speed estimation of the other parameters to be estimated.
5. Method according to one of the preceding claims, characterized in that an estimated value for the range component of the position vector of the respective candidate is determined from the individual range positions of a candidate in the individual single-look SAR images and is kept constant during the optimization method for joint position and speed estimation of the other parameters to be estimated.
6. Method according to one of the preceding claims, characterized in that each candidate that is found during the detection of candidates is allocated the corresponding regions in the respective single-look SAR images as individual initial forms.
7. Method according to one of the preceding claims, characterized in that the form of a candidate is estimated by equalization of the respective initial forms, on the basis of the estimated speed of a candidate.
8. Method according to Claim 6 or 7, characterized in that the imaging errors of the moving objects within a sequence in the single-look SAR images are compensated for by extracting those moving objects with the corresponding initial forms from the signal-look SAR images and by reinserting them at the estimated or calculated positions, by means of the initial forms or the estimated forms, into the single-look SAR images.

Images